I don’t think the tournament rules specify when a game officially starts, though it might be in there.
When time is called, the player whose turn it is should finish it and then the other player takes the final turn. This might be interpreted as a game starting when the corp starts their first turn (so after mulligan etc).
If you rule that a game starts when the previous one ends, and time is called right after the first game ends, then you should draw starting hands and play the first turns (like you said, someone could score points for sure).
Edit: I’m apparently echoing previous posts. Anyway, if it isn’t in the tournament rules, it’s definitely not clear how the case should be handled.